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X and Y are points on the side LN of the...

X and Y are points on the side LN of the triangle LMN such that LX = XY = YN. Through X, a line is drawn parallel to LM to meet MN at Z (see figure). Prove that `ar (DeltaLZY) = ar (MZYX)`.

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Given X and Y are points on the side LN such as that
LX = XY = YN and `XZ || LM`
To prove `" " ar (DeltaLZY) = ar (MZYX)`
Proof Since, `DeltaXMZ` and `DeltaXLZ` are on the same base XZ and between the same parallel lines LM and XZ.
Then, `" " ar (DeltaXMZ) = ar (DeltaXLZ)" "` ...(i)
On adding `ar (DeltaXYZ)` both sides of Eq. (i), we get
`ar (DeltaXMZ) + ar (DeltaXYZ) = ar (DeltaXLZ) + ar (DeltaXYZ)`
`rArr` `" " ar (MZYX) = ar (DeltaLZY)" "` Hence proved.
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